Basic Riemannian Geometry - Department of Mathematical ...

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Definition. A tangent vector at m ∈ M is a linear map ξ : C ∞ (M) → Rsuch thatfor all f, g ∈ C ∞ (M).ξ(fg) = f(m)ξ(g) + ξ(f)g(m)Denote by M m the vector space of all tangent vectors at m.Here are some examples1. For γ : I → M a (smooth) path with γ(t) = m, define γ ′ (t) ∈ M m byγ ′ (t)f = (f ◦ γ) ′ (t).Fact. All ξ ∈ M m are of the form γ ′ (t) for some path γ.2. Let (U, x) be a chart with coordinates x 1 , . . . , x n and x(m) = p ∈ R n .Define ∂ i|m ∈ M m by∂ i|m f = ∂(f ◦ ∣ x−1 ) ∣∣∣p∂x iFact. ∂ 1|m , . . . , ∂ n|m is a basis for M m .3. For p ∈ U ⊂ R n open, we know that U p is canonically isomorphic toR n viafor v ∈ R n .vf = df p (v)4. Let M = f −1 {c} be a regular level set of f : Ω ⊂ R n → R. One canshow that M m is a linear subspace of Ω m∼ = R n . Indeed, under thisidentification,M m = {v ∈ R n : v ⊥ ∇f m }.Now that we have got our hands on tangent vectors, the definition of thederivative of a function as a linear map on tangent vectors is almost tautological:Definition. For f ∈ C ∞ (M), the derivative df m : M m → R of f at m ∈ Mis defined bydf m (ξ) = ξf.4
We note:1. Each df m is a linear map and the Leibniz Rule holds:d(fg) m = g(m)df m + f(m)dg m .2. By construction, this definition coincides with the usual one when Mis an open subset of R n .Exercise. If f is a constant map on a manifold M, show that each df m = 0.The same circle of ideas enable us to differentiate maps between manifolds:Definition. For φ : M → N a smooth map of manifolds, the tangent mapdφ m : M m → N φ(m) at m ∈ M is the linear map defined byfor ξ ∈ M m and f ∈ C ∞ (N).dφ m (ξ)f = ξ(f ◦ φ),Exercise. Prove the chain rule: for φ : M → N and ψ : N → Z andm ∈ M,d(ψ ◦ φ) m = dψ φ(m) ◦ dφ m .Exercise. View R as a manifold (with a single chart!) and let f : M → R.We now have two competing definitions of df m . Show that they coincide.The tangent bundle of M is the disjoint union of the tangent spaces:T M = ∐M m .m∈M1.3 Vector fieldsDefinition. A vector field is a linear map X : C ∞ (M) → C ∞ (M) suchthatX(fg) = f(Xg) + g(Xf).Let Γ(T M) denote the vector space of all vector fields on M.We can view a vector field as a map X : M → T M with X(m) ∈ M m :indeed, we haveX |m ∈ M p5
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Basic Riemannian Geometry - Department of Mathematical ...

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